Parking on a random rooted plane tree

In this paper, we investigate a parking process on uniform random rooted plane tree with $n$ vertices. Every vertex of the has space for single car. Cars arrive at independent uniformly vertices tree. If is unoccupied when car arrives there, it parks. not, drives towards root and parks in first empty encounters (if there one). We are interested asymptotics probability event that all cars can park $\lfloor\alpha n\rfloor$ arrive, $\alpha>0$. observe phase transition $\alpha_{c}:=\sqrt{2}-1$: if $\alpha<\alpha_{c}$ then positive limiting probability, whereas $\alpha>\alpha_{c}$ its tends to 0. Analogous results have been proved by Lackner Panholzer (J. Combin. Theory Ser. A 142 (2016) 1–28), Goldschmidt Przykucki (Combin. Probab. Comput. 28 (2019) 23–45) Jones Appl. 56 1065–1085) different underlying models.